Quantum arithmetic with the Quantum Fourier Transform (1411.5949v2)

Abstract: The Quantum Fourier Transform offers an interesting way to perform arithmetic operations on a quantum computer. We review existing Quantum Fourier Transform adders and multipliers and propose some modifications that extend their capabilities. Among the new circuits, we propose a quantum method to compute the weighted average of a series of inputs in the transform domain.

Quantum Arithmetic with the Quantum Fourier Transform

This paper explores the utilization of the Quantum Fourier Transform (QFT) as a framework for implementing arithmetic operations on quantum computers. The work provides a thorough examination of existing QFT-based adders and multipliers, proposing modifications to extend their capabilities to include both modular and non-modular arithmetic, as well as operations involving signed integers.

Key Contributions

1. QFT-Based Arithmetic Circuits: The paper reviews established QFT circuits for addition and multiplication, such as Draper's QFT adder and its generalizations. These circuits inherently support modular arithmetic, benefiting from the natural alignment of the QFT with modular operations, due to the periodic nature of the quantum phases it manipulates.
2. Non-Modular Arithmetic and Extended Functionality: To accommodate non-modular operations, the authors propose encoding integers into larger quantum registers, ensuring sufficient dimensionality to handle the resultant sums without overflow, and adapting the gates accordingly. This method also facilitates handling signed integers by differentiating positive and negative numbers within the quantum superposition phase space.
3. Implementation and Scalability: The implementation details discussed for QFT adders and multipliers utilize a combination of controlled rotation gates and basic quantum gates, achieving a complexity of $O(n^2)$ for addition and $O(n^3)$ for multiplication, where $n$ is the number of bits representing the integers.
4. Computing Weighted Sums: The paper introduces circuits capable of calculating weighted averages of quantum inputs, leveraging the QFT's capacity to handle phase-based arithmetic efficiently. By modifying the controlled phase shifts introduced by QFT operations, the authors achieve calculations of weighted sums and propose a method for computing inner products of quantum states as a practical application.
5. Quantum Circuit for Weighted Sums: Beyond simple sums, the authors present a novel circuit for computing general weighted sums with quantum weights and values, enabling programmable weighted addition. This circuit supports applications in quantum machine learning, particularly for operations like neural network training that rely on weighted sums.

Implications and Future Directions

The development of efficient QFT-based arithmetic circuits illustrates the potential of quantum computing to expedite complex arithmetic operations. By extending these circuits to support a broader class of operations, the research opens pathways for their integration into more sophisticated quantum algorithms, including those used in quantum machine learning and cryptography.

The implications of these advancements are twofold: on the theoretical side, they provide a clearer understanding of how phase encoding can be harnessed for quantum arithmetic; on the practical side, they offer scalable solutions to arithmetic operations that lie at the heart of various quantum computing applications.

Future research could further expand on these circuits by exploring fault-tolerant versions, considering noise and error impacts on phase coherence, and integrating these operations into more comprehensive quantum algorithms. As quantum hardware continues to advance, the practical implementation of such arithmetic circuits will likely become increasingly relevant, paving the way for their integration into quantum processors dedicated to specific complex computations.